Method of evaluating a physical quantity representing an interaction between a wave and an obstacle

ABSTRACT

The invention relates to the modelling of the interactions between an incident wave and an obstacle, in particular in the area of nondestructive testing. According to the invention, the surface of the obstacle is meshed and at least one source (S i ) is allocated to each surface element (dS i ). Boundary conditions are then calculated at each mesh cell of the obstacle and source values are deduced therefrom. On the basis of an interaction matrix and of these source values, a physical quantity representative of the interaction between the wave and the obstacle is estimated at any point of space.

The invention relates to the modelling of the interactions between anincident wave and an obstacle of this wave, in particular in the area ofnondestructive testing.

The modelling of the interactions between a wave and an obstaclereceiving this wave, such as a target placed in the responsive zone of asensor, finds an advantageous application in nondestructive testing.

A method of modelling called “finite elements” is known consisting inapplying a tiling of the three-dimensional space surrounding theobstacle and in evaluating the aforesaid interactions for all the tilesof the space.

Methods of computation by “finite elements” afford a solution to aproblem posed in the form of partial differential equations. They arebased on a representation of the space under study by an assemblage offinite elements, inside which are defined approximation functionsdetermined in terms of nodal values of the physical quantity sought. Thecontinuous physical problem therefore becomes a discrete finite elementproblem where the nodal quantities are the new unknowns. Such methodstherefore seek to approximate the global solution, rather than thestarting equations in the partial spatial derivatives.

The discretization of the space taken into account ensures that thelatter is entirely covered by finite elements (lines, surfaces orvolumes), this operation is called “meshing” in two dimensional space(2D) or “tiling” in three-dimensional space (3D). The elements involvedare either rectangular or triangular in 2D, or parallelepipedal ortetrahedral in 3D. They may be of different sizes, distributed uniformlyor otherwise over the surface.

In general, the physical quantity sought, such as an electrostaticpotential or a pressure value, is known on the boundary of the domain.This boundary may be fictitious. Boundary conditions are imposed there.The potential is therefore unknown inside the same domain. A node isthen defined as being a vertex of an element. The unknowns of theproblem are therefore the values of the potential at each node of thedomain as a whole.

By way of illustration, FIG. 6 of the prior art represents an exemplarysurface, consisting of two materials M1 and M2, of differentelectromagnetic properties, and meshed by triangular elements eachcomprising three nodes Ai, Bi and Ci. The domain as a whole is delimitedby a boundary F.

Once the mesh has been defined, several approaches exist fortransforming the physical formulation of the problem into a discretemodelling by finite elements. If the problem is formulated throughdifferential equations and consists in minimizing a functional, then avariational procedure is generally applied. This transformation leads toa matrix formulation which when solved gives the nodal solutions, thesolutions at the non-nodal points being obtained by linearinterpolation.

Nevertheless, such computations, in three dimensions, requireconsiderable computing resources and generate very long computationtimes, despite the enhancement in the performance of software allowingthe implementation of these computations.

Admittedly, 2D problems, often simplified by symmetry conditions thatare advantageous for modelling only part of the geometry, are solvedrapidly. However, this is not so for 3D problems, which are the mostfrequent. FIG. 6 shows how the fineness of the mesh, that is to say theratio of the size of an element to that of the smallest detail of thedomain, amplifies the number of nodes.

Consequently, the number of equations and of unknowns increasesproportionally, and, hence, the computation time required for solvingthe problem. It is important to point out that the generation of themesh, namely the discretization of the workspace, and the generation ofthe list of nodes consumes greater computation time than that requiredfor solving the problem.

The present invention aims to improve the situation.

Accordingly it proposes a method of evaluating a physical quantityassociated with an interaction between a wave and an obstacle, in aregion of three-dimensional space, in which:

-   a) a plurality of surface samples, of which a part at least    represents the surface of an obstacle receiving a main wave and    emitting, in response, a secondary wave, is determined by meshing,    and at least one source emitting an elementary wave representing a    contribution to the said secondary wave is allocated to each surface    sample,-   b) a matrix system is formed, comprising:    -   an invertible interaction matrix, applied to a given region of        space and comprising a number of columns corresponding to a        total number of sources,    -   a first column matrix, each coefficient of which is associated        with a source and characterizes the elementary wave that it        emits,    -   and a second column matrix, which is obtained by multiplication        of the first column matrix by the interaction matrix and the        coefficients of which are values of a physical quantity        representative of the wave emitted by the set of sources in the        said given region,-   c) to estimate the coefficients of the first column matrix, chosen    values of physical quantity are assigned to predetermined points,    each associated with a surface sample, the said chosen values being    placed in the second column matrix, and this second column matrix is    multiplied by the inverse of the interaction matrix applied to the    said predetermined points,-   d) to evaluate the said physical quantity representing the wave    emitted by the set of sources in a given region of three-dimensional    space, the interaction matrix is applied to the said given region    and this interaction matrix is multiplied by the first column matrix    comprising the coefficients estimated in step c).

Thus, according to one of the advantages afforded by the presentinvention, the meshing step a) relates only to one or more surfaces,whereas the method of modelling of the “finite element” type requires atiling of the whole space neighbouring the obstacle, thereby making itpossibly to reduce, in the implementation of the method according to theinvention, the memory resources and the computation times required.

The method according to the invention applies equally well to a mainwave emitted by a far source as to a main wave emitted in the nearfield.

Advantageously, to evaluate a physical quantity representative of aninteraction between an element radiating a main wave and an obstaclereceiving this main wave,

-   -   in step a), a plurality of surface samples together representing        an active surface of the element radiating the main wave is        furthermore determined, by meshing, and at least one source        emitting an elementary wave representing a contribution to the        said main wave is allocated to each sample of the active        surface,    -   steps b), c) and d) are applied to the samples of the active        surface, and    -   the said physical quantity representing the interaction between        the radiating element and the obstacle in a given region of        three-dimensional space is evaluated by taking account of the        contribution, in the said given region, of the main wave emitted        by the set of sources of the active surface and the contribution        of the secondary wave emitted by the set of sources of the        surface of the obstacles.

The terms “radiating element” are understood to mean either an emitterof the main wave, such as a wave generator, or a receiver of the mainwave, such as a sensor of this wave.

In a first embodiment, the physical quantity to be evaluated is a scalarquantity and, in step a), a single source is allocated to each surfacesample.

In a second embodiment, the physical quantity to be evaluated is avector quantity expressed by its three coordinates in three-dimensionalspace, and three sources are allocated, in step a), to each surfacesample.

In an advantageous embodiment, to estimate, in step d), the contributionof the secondary wave in the given region of space, the values ofphysical quantity chosen in step c) are dependent on a predeterminedcoefficient of reflection and/or of transmission of the main wave byeach surface sample of the obstacle.

Thus, it will be understood that the secondary wave may eithercorrespond to a reflection of the main wave, or to a transmission of themain wave, or else to a diffraction of the main wave. In thisadvantageous embodiment, step c) corresponds finally to a determinationof the boundary conditions at the surface of the obstacle, in the guiseof interface between two distinct media in particular in aheterostructure.

Furthermore, for nondestructive testing of a target forming an obstacleof a main wave, a chosen coefficient of reflection or of transmission isallocated to all the predetermined points of the surface of the target,and a simulation obtained by the implementation of the method within themeaning of the invention is compared with an experimental measurement.Thus, the points of the surface of the target which, in the experimentalmeasurement, do not satisfy the simulation correspond to inhomogeneitiesor to impurities on the surface of the target.

In another approach, the global properties of the obstacle are known, inparticular in transmission and/or in reflection. By the implementationof the method of the invention, the position in space of a sensor oreven the shape of this sensor is then optimized for application tonondestructive testing, this sensor being intended to analyse a targetforming an obstacle of the main wave.

Accordingly, in an advantageous embodiment, a plurality of values of thephysical quantity estimated in step d) within the meaning of the methodof the invention, which are obtained for a plurality of regions ofspace, are compared so as to select a candidate region for the placementof a radiating element intended to interact with the obstacle.

As indicated hereinabove, the terms “radiating element” are understoodto mean either a sensor or a generator of the wave. It will thus beunderstood that the optimization of the position of the radiatingelement can be applied also to the optimization of the placement or ofthe shape of a wave generator. For example, the present invention alsofinds an advantageous application to the placement of loudspeakers in aclosed volume, delimited by obstacles, such as for example the cabin ofa motor vehicle.

Other characteristics and advantages of the invention will becomeapparent on examining the detailed description hereinbelow, and theappended drawings in which:

FIG. 1A diagrammatically represents the respective surfaces of aradiating element ER emitting a wave and of an obstacle OBS receivingthis wave, said surfaces being meshed with a view to evaluating a scalarquantity representative of the wave at a point M of three-dimensionalspace;

FIG. 1B represents in detail a surface sample dS_(i) corresponding to amesh cell of FIG. 1A, as well as a source S_(i) associated with thesurface sample dS_(i);

FIG. 2A diagrammatically represents the respective surfaces of aradiating element ER emitting a wave and of an obstacle OBS receivingthis wave, said surfaces being meshed with a view to evaluating a vectorquantity representative of the wave at a point M of three-dimensionalspace;

FIG. 2B represents in detail a surface sample dS_(i) corresponding to amesh cell of FIG. 2A, as well as three associated sources SA_(i), SB_(i)and SC_(i);

FIG. 2C represents, viewed face on, a meshed surface, each surfacesample of which comprises three sources SA_(i), SB_(i) and SC_(i), forthe estimation of a vector quantity;

FIG. 3A represents, by way of illustration, the plates of a capacitor,of respective electric potentials V1 and V2, for the estimation of anelectric potential at the point M of three-dimensional space, a singlesource S_(i) being associated with each surface sample dS_(i) of FIG.3A;

FIG. 3B represents, by way of illustration, the plates of a capacitor,of respective electric potentials V1 and V2, for the estimation of anelectric field {right arrow over (E)}(M), at the point M ofthree-dimensional space, three sources SA_(i), SB_(i) and SC_(i) beingassociated with each surface sample dS_(i) of FIG. 3B;

FIG. 4A represents, like FIGS. 1A and 2A, an interaction between aradiating element ER and an obstacle OBS, so as to evaluate a physicalquantity (scalar or vector) at a point M in a portion of space delimitedby the surface of the radiating element and the surface of the obstacle,this point of space M receiving both the wave emitted by the radiatingelement and the wave reflected by the obstacle;

FIG. 4B, complementary to FIG. 4A, represents a transmission by theobstacle OBS of the wave emitted by the radiating element ER, at a pointM of a half-space delimited by the plane formed by the surface of theobstacle OBS;

FIG. 5A diagrammatically represents an obstacle OBS, of finitedimensions, with sources associated with the surface samples andarranged so as to estimate a quantity representative of a reflection ofthe wave off the obstacle;

FIG. 5B, as a supplement to FIG. 5A, diagrammatically represents anobstacle OBS, of finite dimensions, with the sources associated with thesurface samples and placed so as to estimate a quantity representativeof the transmission of the wave by the obstacle;

FIG. 5C represents a simulation of an ultrasound wave emitted by aradiating element ER and propagating towards an obstacle OBS;

FIG. 6 represents a mesh of three-dimensional media, for the applicationof a method of computation by “finite elements”, within the meaning ofthe state of the art;

FIG. 7A represents in detail a surface element and an observation pointM whose relative positions are labelled by an angle θ; and

FIG. 7B diagrammatically represents a surface to be meshed of complexshape, in particular with an observation point M lying in a shadow zonewith respect to certain sources of the surface.

Reference is first made to FIG. 1A, in which the surface of an obstacleOBS, receiving a wave, is meshed according to a plurality of surfacesamples dS₁ to dS₄, in accordance with the aforesaid step a).

Referring to FIG. 1B, with each surface sample dS_(i) is associated ahemisphere HEM_(i), tangential to the surface sample dS_(i) at a pointof contact P_(i). Preferably, this point of contact P_(i) corresponds tothe apex of the hemisphere HEM_(i). For the estimation of a scalarphysical quantity at the point M (such as an electrostatic potential, anacoustic pressure or the like), a single source S_(i) is associated withthe surface sample dS_(i). As will be seen later, in the case of theestimation of a vector quantity in a point of space M, three sourceswill rather be assigned to each surface sample dS_(i).

Preferably, the hemisphere HEM_(i) is constructed as describedhereinbelow. During the aforesaid meshing step a), the surface area ofthe obstacle OBS is on the one hand evaluated, and, on the other hand, anumber of surface samples dS_(i) is chosen according to the desiredaccuracy of estimation of the physical quantity at the point M. Thus,the surface area of a sample dS_(i) is given by S_(o)/N, where S_(o)corresponds to the total surface area of the obstacle and N correspondsto the chosen number of surface samples dS_(i).

The hemisphere HEM_(i) has the same surface area as the sample dS_(i).Thus, the radius R_(i) of the hemisphere is deduced from the expression:${2\quad\pi\quad R_{i}^{2}} = \frac{S_{o}}{N}$

Each mesh cell represented by a surface sample dS_(i) exhibits, in theexample described, a parallelogram shape, whose centre P_(i) correspondsto the point of intersection of the diagonals of this parallelogram. Thehemisphere HEM_(i) is tangential to the surface sample dS_(i) at thispoint P_(i). Of course, the mesh cells may be of different shape,triangular or other. It is indicated generally that the point P_(i)corresponds to the barycentre of the mesh cell.

The position of the source S_(i) (situated at the centre of thehemisphere HEM_(i)) is thus defined. The distance separating the sourceS_(i) from the point of contact P_(i) corresponds to the radius R_(i) ofthe hemisphere HEM_(i) and the straight line which passes through thepoints P_(i) and S_(i) is orthogonal to the mesh cell dS_(i).

In the example represented in FIG. 1A, the surface of a radiatingelement ER, corresponding for example to a wave generator, isfurthermore meshed. With each mesh cell of the surface of the radiatingelement is associated a surface sample dS′_(i), as will be seen later.

The matrix system that is formulated in the aforesaid step b)corresponds to: $\begin{matrix}{\begin{pmatrix}{V\left( M_{1} \right)} \\{V\left( M_{2} \right)} \\\vdots \\{V\left( M_{N} \right)}\end{pmatrix} = {F \times \begin{pmatrix}v_{1} \\v_{2} \\\vdots \\v_{N}\end{pmatrix}}} & \lbrack 1\rbrack\end{matrix}$where:

-   -   the coefficients v_(j) (with j=1, 2, . . . , N) of the first        column matrix correspond to values each associated with a        source, such as an electric charge (in the case of the        estimation of an electric potential), or to a magnetic flux (in        the case of the estimation of a magnetic potential), or else to        a sound speed (in the case of the estimation of an acoustic        pressure related to the propagation of a sound wave);    -   the coefficients V(M_(i)) (with i=1, 2, . . . , N) of the second        column matrix each correspond to a value of the physical        quantity (an electric or magnetic potential or a pressure) to be        estimated at a point M_(i) of space);    -   the interaction matrix F comprises coefficients C_(i,j), the        general expression for which is given by:        C _(i,j) =f(M _(i) S _(j))  [2]

It is thus understood that the coefficients of the matrix F areinteraction coefficients which depend on the distance separating eachpoint of the space M_(i) from a source S_(j) associated with a mesh celldS_(j).

In the case of the propagation of an electric wave, the coefficientsc_(i,j), v_(j) and V(M_(i)), respectively of the interaction matrix ofthe first and of the second column matrix, are given by: $\begin{matrix}{{C_{i,j} = \frac{1}{2\quad\pi\quad ɛ_{0}\overset{\_}{M_{i}S_{j}}}}{v_{j} = q_{j}}{{V\left( M_{i} \right)} = U_{i}}} & \lbrack 3\rbrack\end{matrix}$where:

-   -   ε₀ is a dielectric constant,    -   {overscore (M_(i)S_(j))} is a distance measured as an algebraic        value,    -   q_(j) corresponds to an electric charge characterizing a source        S_(j), and    -   U_(i) corresponds to an electric potential at the point M_(i).

In the case of the propagation of a magnetic wave, the expression forthese coefficients is as follows: $\begin{matrix}{{C_{i,j} = \frac{1}{2\quad\pi\quad\mu_{0}\overset{\_}{M_{i}S_{j}}}}{v_{j} = \varphi_{j}}{{V\left( M_{i} \right)} = \theta_{i}}} & \lbrack 4\rbrack\end{matrix}$where:

-   -   μ₀ corresponds to the magnetic permeability of the medium where        the point M_(i) is situated,    -   φ_(j) corresponds to the magnetic flux associated with the        source S_(j);    -   θ_(i) corresponds to the magnetic potential at the point M_(i).

Within the framework of the propagation of an ultrasound wave, thesecoefficients are given by: $\begin{matrix}{{C_{1,j} = {{- \frac{i\quad\omega\quad\rho}{2\quad\pi}}{\frac{\exp\left( {i\quad{\overset{\rightarrow}{k} \cdot \overset{\_}{M_{1}S_{j}}}} \right)}{\overset{\_}{M_{1}S_{j}}} \cdot {dS}_{j}}}}{v_{j} = \overset{\rightarrow}{v_{j}}}{{V\left( M_{i} \right)} = p_{i}}} & \lbrack 5\rbrack\end{matrix}$in which:

-   -   i²=−1.    -   ω is the angular frequency of the sound wave;    -   ρ is the density of the medium in which the point M_(i) is        situated;    -   the vector {right arrow over (v)}_(j) corresponds to the sound        speed emanating from the source S_(j);    -   {right arrow over (k)} corresponds to the wave vector of the        sound wave; and    -   p_(i) corresponds to the acoustic pressure generated by the        propagation of the ultrasound wave at the point M_(i).

In the expression for the coefficients c_(l,j), the term dS_(j)corresponds to the surface area of the sample associated with the sourceS_(j). Preferably, the meshing of a surface within the meaning of stepa) of the method according to the invention is chosen in such a way thateach mesh cell comprises one and the same surface area dS=dS₁=dS₂= . . .=dS_(j).

It is noted in particular in the expression for the coefficients c_(l,j)that they depend on the scalar product of the wave vector and the vector{right arrow over (M_(i)S_(j))}. Thus, for ultrasound waves, account istaken of a phase shift between the paths which join each source S_(j) toa point of three-dimensional space M, this phase shift being due to adifference in journey length between the rays leaving each source andarriving at the point M (as shown in FIG. 4A). In particular, it will beunderstood that the angle of incidence of such a ray is taken intoaccount in the expression for the coefficients of the interaction matrixF.

Of course, within the framework of the propagation of an electromagneticwave of high frequency, hence of short wavelength, which differs fromthe electrostatic or magnetostatic framework hereinabove, account istaken of the propagation term exp (i{right arrow over (k)}.{right arrowover (r)}) in the expression for the interaction matrix, with respect tothe geometry of the problem to be solved, as within the framework of thepropagation of an ultrasound wave hereinabove (relation [5]).

Thus, the matrix system of equation [1] makes it possible to estimate,on the basis of an interaction matrix F and of a vector comprising thevalues v_(j) associated with the sources S_(j), the coefficients of avector (column matrix) comprising the values of physical quantityV(M_(i)) at the points of space M_(i).

To determine the values of the sources v_(j), the following matrixsystem is applied: $\begin{matrix}{\begin{pmatrix}{V\left( P_{1} \right)} \\{V\left( P_{2} \right)} \\\vdots \\{V\left( P_{N} \right)}\end{pmatrix} = {F \times \begin{pmatrix}v_{1} \\v_{2} \\\vdots \\v_{N}\end{pmatrix}}} & \lbrack 6\rbrack\end{matrix}$where:

-   -   the coefficients of the interaction matrix F are expressed by        with C_(i,j)=f(P_(i)S_(j))    -   the indices i and j correspond respectively to the i^(th) row        and the j^(th) column of the interaction matrix F. This        interaction matrix comprises, for the determination of the        values associated with the sources v_(j), N rows and N columns,        recalling that N is the total number of mesh cells on the        surface of the obstacle;    -   the points P_(i) correspond to the apex of the hemispheres        HEM_(i) of FIG. 1B.

The implementation of step c) of the method within the meaning of thepresent invention corresponds to computing a boundary condition for thepoints P_(i), of known properties, as will be seen later.

The matrix system of equation [6] then becomes: $\begin{matrix}{\begin{pmatrix}v_{1} \\v_{2} \\\vdots \\v_{N}\end{pmatrix} = {F^{- 1} \times \begin{pmatrix}{V\left( P_{1} \right)} \\{V\left( P_{2} \right)} \\\vdots \\{V\left( P_{N} \right)}\end{pmatrix}}} & \lbrack 7\rbrack\end{matrix}$where:

-   -   F⁻¹ corresponds to the inverse of the interaction matrix F; and    -   the values V(P_(i)) are predetermined, as a function of the        aforesaid boundary conditions.

The source values v_(j) are thus determined.

On the basis of the estimation of these source values v_(j), it ispossible to compute the scalar physical quantity at any point M ofthree-dimensional space, on the basis of the relation: $\begin{matrix}{{V(M)} = {\sum\limits_{j = 1}^{N}\quad{{f\left( {MS}_{j} \right)} \cdot v_{j}}}} & \lbrack 8\rbrack\end{matrix}$

To obtain this expression for the scalar quantity V(M), the interactionmatrix F may comprise just one row of coefficients c_(j), with:C _(j) =f(MS _(j)),but always comprises N columns.

Referring again to FIG. 1A, it will be understood that the surface ofthe obstacle OBS receiving the wave emitted by the radiating element ERacts, itself, as an active surface re-emitting a secondary wave (forexample by reflection). Each source S_(i) represents a contribution tothe emission of this secondary wave.

Furthermore, to take account both of the presence of the main wave andof the presence of the secondary wave at the point M, the contributionof the main wave and the contribution of the secondary wave at the pointM are estimated via the matrix system: $\begin{matrix}{{V(M)} = {{F \times \begin{pmatrix}v_{1} \\v_{2} \\\vdots \\v_{N}\end{pmatrix}} + {F^{\prime} \times \begin{pmatrix}v_{1}^{\prime} \\v_{2}^{\prime} \\\vdots \\v_{N^{\prime}}^{\prime}\end{pmatrix}}}} & \lbrack 9\rbrack\end{matrix}$where:

-   -   F′ is the matrix of interaction between the surface of the        radiating element and the point M;    -   v′_(j) (with j=1, 2, 3, . . . , N′) is the value of the sources        allocated to each surface sample dS′_(j) of the radiating        element, N′ being the chosen total number of mesh cells of the        active surface of the radiating element ER.

The coefficients of the matrix F′ are again dependent on the distanceMS′_(j), where S′_(j) are the sources assigned to each surface sampledS′_(j) of the radiating element.

According to an advantageous characteristic, the values of the sourcesof the obstacle v_(j) are determined as a function of the values of thesources of the radiating element v′_(j), and which are themselvescomputed as will be seen later with reference to FIGS. 4A, 4B, 5A and5B.

Reference is now made to FIG. 2A, in which three sources are assigned toeach surface sample dS_(i), with a view to estimating a vector physicalquantity {right arrow over (V(M))}, at a point M of three-dimensionalspace.

It will be understood that to estimate the vector quantity, via itsthree coordinates in space x, y and z, the number of equations to besolved with respect to the previous matrix system must be multiplied bythree. Thus, the matrix F⁻¹ of relation [7] must comprise three times asmany rows as before. The interaction matrix F must, itself, comprisethree times as many columns as before and, accordingly, three sourcesare advantageously envisaged per mesh cell when dealing with thedetermination of the coordinates in three-dimensional space of a vector{right arrow over (V(M))}.

Referring to FIG. 2B, the three sources SA_(i), SB_(i), SC_(i),allocated to a surface sample dS_(i) have respective positionsdetermined as is indicated hereinbelow. Such as represented in FIG. 2B,the three sources SA_(i), SB_(i), SC_(i) are coplanar and the planewhich includes these three sources also includes the base of thehemisphere HEM_(i). The hemisphere HEM_(i) is constructed as indicatedhereinabove (with the same surface area as the surface area of the meshcell), with however the centre of the hemisphere which corresponds hereto the barycentre of the three sources SA_(i), SB_(i) and SC_(i).

The “centre of the hemisphere” is understood to mean the centre of thedisk which constitutes the base of the hemisphere.

The three sources which are allocated to the surface sample dS_(i) areplaced at the vertices of an equilateral triangle whose barycentre G_(i)corresponds to the centre of the hemisphere. Preferably, each sourceSA_(i), SB_(i) and SC_(i) is placed in the middle of a radius R_(i) ofthe hemisphere. Thus, the straight lines which connect the barycentreG_(i) to each source are angularly separated by 120°.

Referring to FIG. 2C, the angular orientation of the triangles formed bythe triplets of sources is chosen randomly, from one surface sample toanother. Advantageously, overperiodicity artefacts, which could resultfrom the choosing of one and the same angular orientation of thesetriangles, are thus avoided in the estimation of the vector quantity atthe point M.

With reference to the various wave types indicated previously, thevector quantity {right arrow over (V)}(M) to be estimated may be:

-   -   an electric field, within the framework of the propagation of an        electric wave;    -   a magnetic field, within the framework of the propagation of a        magnetic wave; and    -   a speed of sound at the point M, within the framework of the        propagation of ultrasound waves.

To determine the values associated with each source SA_(i), SB_(i),SC_(i), the matrix system is fomulated according to the followingrelation: $\begin{matrix}{\begin{pmatrix}{V_{x}\left( P_{1} \right)} \\{V_{x}\left( P_{2} \right)} \\\vdots \\{V_{x}\left( P_{N} \right)} \\{V_{y}\left( P_{1} \right)} \\{V_{y}\left( P_{2} \right)} \\\vdots \\{V_{y}\left( P_{N} \right)} \\{V_{z}\left( P_{1} \right)} \\{V_{z}\left( P_{2} \right)} \\\vdots \\{V_{z}\left( P_{N} \right)}\end{pmatrix} = {F_{\overset{\_}{V}} \times \begin{pmatrix}{vA}_{1} \\{vA}_{2} \\\vdots \\{vA}_{N} \\{vB}_{1} \\{vB}_{2} \\\vdots \\{vB}_{N} \\{vC}_{1} \\{vC}_{2} \\\vdots \\{vC}_{N}\end{pmatrix}}} & \lbrack 11\rbrack\end{matrix}$

It is noted, in particular, that the interaction matrix F{right arrowover (V)} is of dimensions 3N×3N, where N is the total number of surfacesamples. The interaction matrix is expressed here through the relation:$\begin{matrix}{F_{\quad\overset{\rightarrow}{V}} = \begin{pmatrix}\underset{\underset{N{\{{C_{A}^{x}{({i,j})}}}}{︷}}{N} & \underset{\underset{C_{B}^{x}{({i,j})}}{︷}}{N} & \underset{\underset{C_{C}^{x}{({i,j})}}{︷}}{N} \\{N\left\{ {C_{A}^{y}\left( {i,j} \right)} \right.} & {C_{B}^{y}\left( {i,j} \right)} & {C_{C}^{y}\left( {i,j} \right)} \\{N\left\{ {C_{A}^{z}\left( {i,j} \right)} \right.} & {C_{B}^{z}\left( {i,j} \right)} & {C_{C}^{z}\left( {i,j} \right)}\end{pmatrix}} & \lbrack 12\rbrack\end{matrix}$

The coefficients of this matrix are expressed through:C _(σ) ^(u)(i,j)=f _(u) [d(P _(i) ,Sσ _(j))]  [13]with σ=A, B, C

-   -   i=1, 2, . . . , N    -   j=1, 2, . . . , N    -   u=x, y, z.        and are also dependent on a distance separating the point of        contact P_(i) from one of the sources Sσ_(j) (σ=A, B or C) of a        triplet associated with a surface sample dS_(j).

By inverting the interaction matrix F_({right arrow over (V)}), thevalues vσ_(j) associated with each source Sσ_(j) are thus determined byapplying boundary conditions on the values of the vector {right arrowover (V)} at the points P_(i). These boundary conditions impose a valueof the vector {right arrow over (V)}, according to its three coordinatesV_(x)(P_(i)), V_(y)(P_(i)) and V_(z)(P_(i)).

Once these source values vσ_(j) have thus been determined, theexpression for the vector {right arrow over (V)} at any point M of spacecan easily be computed through the relation: $\begin{matrix}{\quad{{{\overset{\rightarrow}{V}(M)} = {{{V_{x}(M)}\overset{\rightarrow}{x}} + {{V_{y}(M)}\overset{\rightarrow}{y}} + {{V_{z}(M)}\overset{\rightarrow}{z}}}}{{V_{x}(M)} = {\sum\limits_{\underset{{\sigma = A},B,C}{{j = 1},\ldots\quad,N}}^{\quad}{{{f_{x}\left\lbrack \left( {d\left( {M,{S\quad\sigma_{j}}} \right)} \right) \right\rbrack} \cdot v}\quad\sigma_{j}}}}{{V_{y}(M)} = {\sum\limits_{\underset{{\sigma = A},B,C}{{j = 1},\ldots\quad,N}}^{\quad}{{{f_{y}\left\lbrack \left( {d\left( {M,{S\quad\sigma_{j}}} \right)} \right) \right\rbrack} \cdot v}\quad\sigma_{j}}}}{{V_{z}(M)} = {\sum\limits_{\underset{{\sigma = A},B,C}{{j = 1},\ldots\quad,N}}^{\quad}{{{f_{z}\left\lbrack \left( {d\left( {M,{S\quad\sigma_{j}}} \right)} \right) \right\rbrack} \cdot v}\quad\sigma_{j}}}}}} & \lbrack 14\rbrack\end{matrix}${right arrow over (x)}, {right arrow over (y)} and {right arrow over(z)} correspond to unit vectors plotted along the axes x, y and z ofthree-dimensional space.

Thus, the interaction matrix F_({right arrow over (V)}), when it isapplied to any point M of space, ultimately comprises only three rowseach associated with a coordinate of space x, y or z.

For various types of waves, the values of the sources vσ_(j) are, asbefore, an electric charge in respect of an electric wave, a magneticflux in respect of a magnetic wave, a speed of sound in respect of anultrasound wave.

More precisely, the coefficients of the interaction matrixF_({right arrow over (V)}) are determined from the above relations [3],[4] and [5], specifying however that:{right arrow over (V)}(M)=−{overscore (grad)}[V(M)]  [15]V(M) being the scalar quantity computed previously through equation [8].

Thus, for the estimation of a vector quantity {right arrow over (V)} atthe point M and for the wave types mentioned above by way of example(electric, magnetic and ultrasound), the coefficients of the interactionmatrix F_({right arrow over (V)}) are inversely proportional to thesquare of a distance separating each source from the point M, while forthe estimation of a scalar quantity V at a point M of space, thecoefficients of the interaction matrix F are simply inverselyproportional to this distance. Each distance involves one of the sourcesof a triplet of a surface sample and a point M of space. The interactionmatrix F_({right arrow over (V)}) then comprises 3N columns when takingthree sources per surface sample, while the interaction matrix F for theestimation of the scalar quantity comprised only N columns since justone source per surface sample was necessary.

More generally, one source per sample is allocated when boundaryconditions are known for a scalar quantity and three sources per sampleare allocated when boundary conditions are in fact known for a vectorquantity.

Reference is now made to FIG. 3A to describe, by way of illustration, anapplication of the method according to the invention to the estimationof an electric potential at a point M of three-dimensional space,situated between two plates of a capacitor. The plates of this capacitorare brought to respective potentials V1 and V2. The implementation ofstep a) consists firstly in meshing the respective surfaces of the twoplates. In the example represented in FIG. 3A, only two mesh cells havebeen represented for each plate, simply by way of illustration.

The application of step b) consists in formulating the matrix systeminvolving the interaction matrix F and the column vector comprising thevalues of the sources S₁ to S₄. Multiplication of these two matricesmakes it possible to obtain a column vector comprising the values of thepotential at one or more points M of space.

The implementation of step c) of the method according to the inventionconsists in applying the matrix system to the points of contact of thehemispheres P₁ to P₄, of each surface sample dS₁ to dS₄. This results inthe following relation: $\begin{matrix}{\begin{pmatrix}{V\left( P_{1} \right)} \\{V\left( P_{2} \right)} \\{V\left( P_{3} \right)} \\{V\left( P_{4} \right)}\end{pmatrix} = {\frac{1}{2\quad\pi\quad ɛ_{0}}\begin{pmatrix}\frac{1}{P_{1}S_{1}} & \frac{1}{P_{1}S_{2}} & \frac{1}{P_{1}S_{3}} & \frac{1}{P_{1}S_{4}} \\\frac{1}{P_{2}S_{1}} & \frac{1}{P_{2}S_{2}} & \frac{1}{P_{2}S_{3}} & \frac{1}{P_{2}S_{4}} \\\frac{1}{P_{3}S_{1}} & \frac{1}{P_{3}S_{21}} & \frac{1}{P_{3}S_{3}} & \frac{1}{P_{3}S_{4}} \\\frac{1}{P_{4}S_{1}} & \frac{1}{P_{4}S_{2}} & \frac{1}{P_{4}S_{3}} & \frac{1}{P_{4}S_{4}}\end{pmatrix}\begin{pmatrix}v_{1} \\v_{2} \\v_{3} \\v_{4}\end{pmatrix}}} & \lbrack 16\rbrack\end{matrix}$ $\begin{matrix}{{\begin{pmatrix}{V\left( P_{1} \right)} \\{V\left( P_{2} \right)} \\{V\left( P_{3} \right)} \\{V\left( P_{4} \right)}\end{pmatrix} = {\frac{1}{2\pi\quad ɛ_{0}}\begin{pmatrix}\frac{1}{P_{1}S_{1}} & \frac{1}{P_{1}S_{1}} & \frac{1}{P_{1}S_{3}} & \frac{1}{P_{1}S_{4}} \\\frac{1}{P_{2}S_{1}} & \frac{1}{P_{2}S_{2}} & \frac{1}{P_{2}S_{3}} & \frac{1}{P_{2}S_{4}} \\\frac{1}{P_{3}S_{1}} & \frac{1}{P_{3}S_{21}} & \frac{1}{P_{3}S_{3}} & \frac{1}{P_{3}S_{4}} \\\frac{1}{P_{4}S_{1}} & \frac{1}{P_{4}S_{2}} & \frac{1}{P_{4}S_{3}} & \frac{1}{P_{4}S_{4}}\end{pmatrix}\begin{pmatrix}v_{1} \\v_{2} \\v_{3} \\v_{4}\end{pmatrix}}}{with}\quad{{V\left( P_{1} \right)} = {{V\left( P_{2} \right)} = V_{1}}}{{V\left( P_{3} \right)} = {{V\left( P_{4} \right)} = V_{2}}}{{V_{1} = q_{1}},{v_{2} = q_{2}},{v_{3} = q_{3}},{v_{4} = q_{4}}}} & \lbrack 16\rbrack\end{matrix}$

Here, the boundary condition prescribes that the value of the potentialat the points of contact P₁ and P₂ should correspond to the potential V1of the first plate. Likewise, the electric potential at the points ofcontact P₃ and P₄ should correspond to the electric potential of thesecond plate V2. By inverting the interaction matrix applied to thepoint of contact P_(i), the values of the sources v_(i) whichcorrespond, as expressed in relation [16], to electric charges q_(i) aredetermined.

The coefficients of the interaction matrix$\frac{1}{2{\pi ɛ}_{0}P_{i}S_{j}}$are known perfectly, since the positions of the sources S_(j) and thepositions of the points of contact P_(i) are determined beforehand, asis represented in FIG. 1B.

The expression for the electric potential V(M) at the point M betweenthe two plates is ultimately given by the expression: $\begin{matrix}{{V(M)} = {\frac{1}{2\quad\pi\quad ɛ_{0}}\left( {\frac{q_{1}}{{MS}_{1}} + \frac{q_{2}}{{MS}_{2}} + \frac{q_{3}}{{MS}_{3}} + \frac{q_{4}}{{MS}_{4}}} \right)}} & \lbrack 17\rbrack\end{matrix}$Reference is now made to FIG. 3B in which the same plates have beenrepresented as in FIG. 3A, with substantially the same mesh, but withthe aim, here, of estimating a vector quantity corresponding to theelectric field {right arrow over (E(M))}, at the point M ofthree-dimensional space.

Relations [11] to [15] can be applied to estimate the value of theelectric field at the point M, with, in relation [13]: $\begin{matrix}{{C_{\sigma}^{\mu}\left( {i,j} \right)} = \left( \frac{1}{2\quad\pi\quad ɛ_{0}{d^{2}\left( {P_{i},{S\quad\sigma_{j}}} \right)}} \right)_{u}} & \lbrack 18\rbrack\end{matrix}$with

-   -   σ=A, B, C    -   u=x, y, z    -   i=1, 2, 3, 4    -   j=1, 2, 3, 4.

However, the values of the electric field at the point of contact P_(i)remain to be determined in relation [11].

A predetermined general law for the behaviour of the field (inreflection, in transmission or other) at the level of the surface of theobstacle (plates in the example of the aforesaid capacitor) is thenintroduced to ascertain the values of the sources vσ_(j).

For example, if the electric wave is reflected totally by the surface ofan obstacle (for example one of the two plates), the electric field at apoint of contact P_(i) is normal to the surface dS_(i) and itscomponents E_(x) and E_(y) are zero. By way of illustration, if thesurface of the plate was represented only by a single surface samplewith three sources, the values of its sources vA, vB and vC would all bemutually equal to one and the same value +q.

On the other hand, if the coefficient of reflection is practically zeroat the surface dS_(i), the component of the electric field E_(z) at thepoint P_(i) is zero, this corresponding indeed to the case where thefield is substantially tangential to the surface dS_(i). Thus, by way ofillustration, if the surface of the plate was represented only by asingle surface sample with three sources, the values of its sources vA,vB and vC would be, for example, +q, +q and −2q respectively. Forexample, within the framework of the propagation of a magnetic wave, ifthe surface of an eddy current sensor (with a zero normal component ofthe magnetic field) was represented by a single surface sample, themagnetic fluxes of the three sources associated with this surface samplewould be +φ, +φand −2φ.

It is thus understood that with the three sources per sample dS_(i), itis possible to define, for example as a function of the weighting ofeach source, any orientation of the field at the surface of theobstacle.

Of course, this approach assumes that the coefficient of reflection R ofan obstacle is known beforehand. In particular, it may be advantageousto compare a simulation and an experimental measurement so as to detect,at the surface of an obstacle, inhomogeneities or impurities whichcorrespond to points of the surface of this obstacle which do notsatisfy the values of the coefficient of reflection R that areprescribed at each predetermined point P_(i) of the obstacle.

A predetermined value of the reflection coefficient can thus be assignedto each point P_(i) of the surface of the obstacle. Accordingly, amatrix R which is representative of the reflection coefficient at eachpoint P_(i) is introduced. For an interaction between a radiatingelement and an obstacle, it is thus possible to express the matrixsystem of relation [9] in another way, that is to say by giving a singleexpression for all the sources of the system (both of the obstacle andof the radiating element), as indicated hereinbelow.

In what follows, it is indicated that:

-   -   F(P) is the interaction matrix of the obstacle OBS, applied to        the points P_(i) of the surface of the obstacle OBS;    -   F(P′) is the interaction matrix of the obstacle OBS, applied to        the points P′i of the surface of the radiating element ER;    -   F′ (P) corresponds to the interaction matrix of the radiating        element ER, applied to the points P_(i) of the surface of the        obstacle OBS;    -   F′(P′) corresponds to the interaction matrix of the radiating        element ER, applied to the points P_(i) of the surface of the        radiating element ER;    -   {right arrow over (v)}′ corresponds to the column vector        comprising the values of the sources S′_(i) of the radiating        element ER; and    -   {right arrow over (v)} corresponds to the column vector        comprising the values of the sources S_(i) of the obstacle OBS.

On an obstacle, the contribution of the wave emitted by the radiatingelement ER is expressed by:{right arrow over (V)}′(P)=F′(P).{right arrow over (v)}′  [19]

The contribution of the secondary wave returned by the obstacle OBS isexpressed, by definition, by the relation:{right arrow over (V)}(P)=F(P).{right arrow over (v)}  [20]

Now, in the example represented in FIG. 4A, the secondary wavecorresponds simply to a reflection of the main wave. This is expressedby the relation:{right arrow over (V)}(P)=R{right arrow over (V)}′(P)  [21]where R corresponds to a reflection matrix each coefficient of whichrepresents the contribution to the emission, by reflection, of thesecondary wave, by each source S_(i) (or Sσ_(i), within the framework ofestimation of a vector quantity) of the obstacle OBS.

From the three relations [19], [20] and [21] is deduced the expressionfor the column vector {right arrow over (v)} comprising the values ofthe sources of the obstacle, on the basis of the column vector {rightarrow over (v)} comprising the values of the sources of the radiatingelement, through the relation:{right arrow over (v)}=[F(P)]⁻¹ .R.[F′(P)].{right arrow over (v)}′  [22]

Additionally, for fine estimation of the scalar or vector quantities atthe point M, in particular to take account of multiple reflections, itis advantageous to take account of the contribution of the radiation bythe obstacle, at the level of the surface of the radiating element ER.Accordingly, account is taken, in the estimation of the boundaryconditions at the surface of the radiating element ER (at the pointsP′_(i)) of the contribution of the radiation of the sources S′_(i) ofthe radiating element and of the contribution of the emission of thesecondary wave by the sources S_(i) of the obstacle, through therelation:{right arrow over (V)} _(T)(P′)=F(P′){right arrow over (v)}+F′(P′){rightarrow over (v)}′  [23]

The source values S′_(i) of the radiating element ER can thus betailored, by virtue of relation [23], by taking account of thereflection of the obstacle OBS, according to the following relation:{right arrow over (V)} _(T)(P′)={F(P′).[F(P)]⁻¹ .R.[F′(P)]+F′(P′)}.{right arrow over (v)}′  [10]Thus, boundary conditions are simply prescribed for the radiatingelement, so as to deduce therefrom the values of the sources v′_(i). Inpractice, one will preferably proceed as follows:

after meshing the surfaces, one determines the position of the pointsP_(i) and P′_(i) and of the sources S_(i) and S′_(i);

as a function of the type of wave involved, one determines thecoefficients of the matrices F(P), F′(P), F(P′) and F′(P′);

as a function of a law of reflection of the obstacle, one determines thecoefficients of the reflection matrix as in the example given later inrespect of an ultrasound wave;

as a function of boundary conditions on the radiating element (thebehaviour of which is generally known for a given problem), onedetermines the values of the vector {right arrow over (V)}_(T)(P′) atthe points P′_(i) of the surface of the radiating element and onededuces therefrom the values of the sources S′_(i) of the radiatingelement by inverting relation [10];

one also deduces therefrom the values of the sources S_(i) of theobstacle by applying relation [22];

once the values of all the sources S_(i) and S′_(i) have beendetermined, one can apply the matrix system given by relation [9] to anypoint M of space, by applying the interaction matrices F and F′(involving the position of the point M and the positions of therespective sources S_(i) and S′_(i)) to this point M.

Referring again to FIG. 4A, it is considered that the obstacle OBSrepresents simply an interface between two media M1 and M2, thus forminga dioptric member which may be plane, such as represented in the examplein FIG. 4A, but also curved or of any general shape. The reflectioncoefficients R_(i) associated with each point P_(i) depend, within theframework of the propagation of an ultrasound or electromagnetic wave ofhigh frequency, on the angle of incidence β_(i) of the ray emanatingfrom the source S_(i), at the point of three-dimensional space M.

For an ultrasound wave, the expression for the coefficients ofreflection R_(i) is given by: $\begin{matrix}{R_{i} = \frac{{\rho_{2}c_{2}\cos\quad\beta_{i}} - {\rho_{1}{c_{1}\left\lbrack {1 - \frac{c_{2}^{2}}{c_{1}^{2}} + {\frac{c_{2}^{2}}{c_{1}^{2}}\quad\cos^{2}\beta_{i}}} \right\rbrack}^{1/2}}}{{\rho_{2}c_{2}\cos\quad\beta_{i}} + {\rho_{1}{c_{1}\left\lbrack {1 - \frac{c_{2}^{2}}{c_{1}^{2}} + {\frac{c_{2}^{2}}{c_{1}^{2}}\quad\cos^{2}\beta_{i}}} \right\rbrack}^{1/2}}}} & \lbrack 24\rbrack\end{matrix}$where:

-   -   c₁ is the speed of sound in medium M₁;    -   c₂ is the speed of sound in medium M₂;    -   ρ_(i) is the density of medium M₁;    -   ρ₂ is the density of medium M₂.

In this expression [24], the term cosβ_(i) may simply be estimated as afunction of the coordinates in space of the point M and of the pointrepresenting the source S_(i).

Referring now to FIG. 4B, the same estimation may be undertaken for apoint M situated in the medium M₂. In this case, the wave that the pointM receives is a wave transmitted by the obstacle OBS. In particular, itis noted that the sources of the radiating element ER are no longeractive, owing to the occultation of the radiating element ER by theobstacle OBS. In transmission, the reasoning applies as above with aboundary condition prescribed at the points P_(i) through the values ofthe transmission coefficients T_(i) associated with each point P_(i).Within the framework of the propagation of an ultrasound wave, eachtransmission coefficient T_(i) is given by the relation: $\begin{matrix}{T_{i} = \frac{2\quad\rho_{2}c_{2}\quad\cos\quad\beta_{i}}{{\rho_{2}c_{2}\cos\quad\beta_{i}} - {\rho_{1}{c_{1}\left\lbrack {1 - \frac{c_{2}^{2}}{c_{1}^{2}} + {\frac{c_{2}^{2}}{c_{1}^{2}}\quad\cos^{2}\beta_{i}}} \right\rbrack}^{1/2}}}} & \lbrack 25\rbrack\end{matrix}$

As indicated above, the terms cosβ_(i) may be determined as a functionof the respective coordinates of the sources S_(i) and of the point M.

To estimate the values of sources S_(i) of the obstacle OBS, relation[22] is applied while replacing, however, the reflection matrix R by thetransmission matrix T:{right arrow over (v)}=[F(P)]⁻¹ T[F′(P)]{right arrow over (v)}′  [26]

Within the framework of the propagation of an ultrasound wave, thecoefficients of the matrices R and T are estimated for each source S_(i)and for each point P_(i). In particular, each coefficient T_(i,j) orR_(i,j) of the matrix T or of the matrix R (where i corresponds to thei^(th) row and j corresponds to the j^(th) column) is expressed as afunction of an angle β_(i,j) between a normal to the surface of theobstacle at the point P_(i) and a straight line passing through thepoint P_(i) and through a source S_(j). It is thus possible to write, ina general manner, the two relations expressing the values of thecoefficients of the matrices T and R by the following respectiverelations:T _(i,j) =f _(t) (cosβ_(ij))  [27]R _(i,j) =f _(r)(cos β_(ij))  [28]where f_(t) is given by relation [25] and f_(r) is given by relation[24].

More generally, with reference to FIGS. 4A and 4B, it is indicated that,if the obstacle is considered to be a solid material representing amedium M2 distinct from a medium M1 in which the main wave propagatedinitially:

for a reflection of the main wave off the obstacle in the guise ofmedium M2 (the surface of the obstacle forming a dioptric member betweenthe media M1 and M2), the hemispheres HEM_(i) are oriented outwards fromthe obstacle (FIG. 4A);

for a transmission of the main wave within the obstacle, the hemispheresHEM_(i) are oriented inwards into the obstacle (FIG. 4B).

Reference is now made to FIG. 5A to describe the case of a planeobstacle OBS of finite dimensions, excited by a radiating element ER,inclined by a predetermined angle with respect to the obstacle OBS. Asindicated above, for an ultrasound wave, the inclination of theradiating element will be taken into account to compute the contributionof the wave emitted by the radiating element at the point M.Additionally, in a particularly advantageous manner, a surface whichencompasses the surface of the obstacle is meshed (FIG. 5A). For a sliceof space that is delimited by the radiating element, on the one hand,and the obstacle, on the other hand (FIG. 5A), three types of sourcescan be considered:

-   -   the sources S′_(i) of the radiating element ER,    -   sources SO_(i), which return the secondary wave, by reflection        from the obstacle OBS, as a function of a certain coefficient of        reflection R of the obstacle; and    -   sources SS_(i), which do not return any secondary wave and to        which a zero coefficient of reflection may be allocated if the        obstacle separates two media of identical indices. In this case,        these sources SS_(i) are considered to be “off” in the aforesaid        slice of space and are not taken into account in the        computations of the physical quantity at the point M of FIG. 5A.        On the other hand, these sources SS_(i) may be active through        reflection of the main wave if the obstacle OBS separates two        media of different indices.

Additionally, to estimate the scalar or vector quantities associatedwith a point M of a half-space delimited by the surface encompassing theobstacle OBS (on the right of FIG. 5B), one considers:

-   -   the sources SO'_(i) of the obstacle, which are active through        transmission of the main wave, and    -   the sources SS_(i), to which one now assigns a coefficient of        transmission equal to 1 if the obstacle separates two media of        like indices. These sources SS_(i) behave ultimately (to within        angles of incidence) like the sources S′_(i) of the radiating        element ER.

The sources S′_(i) of the radiating element may then be “off” for thecomputation of the physical quantities in this half-space.

To compute the values {right arrow over (v)}′ of the sources S′_(i) ofthe radiating element (from which are deduced the values {right arrowover (v)} of the sources of the obstacle according to relations [22] and[26]), boundary conditions will simply be applied to the points of theactive surface of the radiating element ER. For example, for ultrasoundwave propagation, it may be indicated that the sound velocities at thepoints of the surface of the radiating element ER are perpendicular tothis surface and have mutually equal moduli v₀.

Generally, it is indicated that the three-dimensional space may thus bedivided up by interfaces delimiting media of distinct properties, eachinterface representing an obstacle within the meaning of the presentinvention. The physical quantities are then computed in each slice ofspace. For example, within the framework of the study of aheterostructure (for several interfaces), the above method may beapplied in respect of successive slices of space by considering twointerfaces: one representing a “radiating element”, within the meaningof FIGS. 4A and 5A, for example through transmission of a received wave,and the other representing an obstacle receiving the transmitted wave.Advantageously, account is taken, for each slice of space, of thecontributions of all the interfaces, as expressed by relations [10] and[22].

However, in a preferred practical embodiment, in particular in order toprogramme the simulation of an interaction, one will advantageouslyconsider all the obstacles of the entire space around a point M and acondition will be prescribed regarding the position of the point M withrespect to each source present in the space.

Preferably, with reference to FIG. 7A, the scalar product {right arrowover (SM)}.{right arrow over (r)} is tested at each iteration withrespect to a source S, for example in the form:$\frac{\quad{\overset{\rightarrow}{SM} \cdot \overset{\rightarrow}{r}}}{{\overset{\rightarrow}{r}} \cdot {\overset{\rightarrow}{SM}}} = {\cos\quad\theta}$where {right arrow over (r)} is the vector connecting the source S tothe point of contact P of the half-sphere with the surface element dSconsidered, in the case where just one source per hemisphere isenvisaged. In the case where a triplet of sources S1, S2, S3 is in factenvisaged, the base of the vector {right arrow over (r)} is preferablysituated at the barycentre of the three sources S1, S2, S3. Moreover, inthe case of three sources per surface sample, the computation of thescalar product concerns each source S_(i) of the triplet S1, S2, S3.

Typically, if cos θ is positive, one takes into account the contributionof this source S in the estimation of the interaction.

On the other hand, if cos θ is negative, a zero (scalar or vector) valueis assigned to this source S in the estimation of the interaction.

As a variant of the computation of the scalar product above, an“altitude” of the point M can be computed. In this case, the testpertains to a quantity of the type: $\begin{matrix}\frac{\quad{\overset{\rightarrow}{SM} \cdot \overset{\rightarrow}{r}}}{\overset{\rightarrow}{r}} & \quad\end{matrix}$Of course, other types of test are possible. For example, in the case ofa computation with respect to the angle θ, this angle can be chosenwithin a cone of chosen aperture, or otherwise.

Ultimately, this approach advantageously makes it possible tosystematize any configuration of the sources with respect to theobservation point M, by simply introducing an extra step for testing, ateach iteration on a source S, the position of this source S with respectto the point M, as indicated above.

This approach proves to be particularly advantageous for surfaces to bemeshed which are relatively complex, in particular when the observationpoint M is liable to be situated in a shadow zone with respect tocertain sources, as shown in FIG. 7B. In this FIG. 7B, the half-spheresassociated with the sources in the shadow zone of the observation pointM and for which, consequently, the contribution is fixed as being zeroin the estimated interaction have been represented by dashes. Of course,in the case where a source is screened by a sampled surface, althoughthe scalar product {right arrow over (SM)}.{right arrow over (r)}associated with this source remains positive, a second test determineswhether the vector {right arrow over (SM)} does or not does cross asampled surface. If it does, this source is considered to be inactivespecifically for the region of the point M.

Thus, in more general terms, the method within the meaning of theinvention preferably envisages at least one extra step, for each surfacesample, for testing the value of a scalar product of:

a first vector {right arrow over (r)} normal to the surface sample anddirected towards the apex P of the hemisphere, such as represented inFIG. 7A, and

a second vector {right arrow over (SM)} drawn between a source Sassociated with this hemisphere and the point M which is situated in theregion of observation, while distinguishing, in particular:

-   -   the case where this scalar product is less than a predetermined        threshold and the contribution of this source is not taken into        account, and    -   the case where this scalar product is greater than a        predetermined threshold and the contribution of this source is        actually taken into account.

In the example above where the angle θ between these two vectors isconsidered, the aforesaid predetermined threshold is of course the valuezero and one simply distinguishes between the cases where the scalarproduct is positive or negative.

Of course, this choice is not limiting so that, for a heterostructurewith several parallel dioptric members, it will again be possible toconsider, advantageously, successive half-spaces, as described abovewith reference to FIGS. 4A and 4B.

The simulation of FIG. 5C corresponds, for an ultrasound wave, to thesituation of FIGS. 5A and 5B by taking account:

-   -   of the contribution of the emission of the main wave by the        radiating element ER;    -   of the contribution of the reflection of this main wave by the        obstacle; and    -   of the contribution of the transmission of the main wave by the        obstacle.

The level lines of FIG. 5C correspond to various magnitudes of acousticpressure. The radiating element ER is placed 10 mm from the obstacle OBSand inclined by 20° with respect to the latter. Interference fringes arenoted in particular in a zone between the obstacle OBS and the radiatingelement ER. Such a simulation may advantageously indicate an idealposition of an ultrasound sensor. These ultrasound sensors customarilycomprise a transducer as active radiating element and a detector formeasuring the ultrasound waves received. The simulation of FIG. 5C maythus moreover indicate the ideal shape of an ultrasound sensor,depending on the desired applications, for a given shape of obstacle.

The simulation of FIG. 5C has been performed by virtue of a matrixcomputation programmed with the aid of the MATLAB© computation software.The chosen total number of mesh cells for the obstacle and for theradiating element (here, a few hundred in all) is then optimized:

-   -   on the one hand, to limit the duration of the computations; and    -   on the other hand, so that the size of the mesh cells remains        less than half a wavelength, so as to satisfy the Rayleigh        criterion.

It is indicated however that, as the elements to be meshed in theimplementation of the method according to the invention are simplysurfaces, the computation times are not nearly as long as those requiredin the implementation of a method of computation of “finite element”type.

The present invention can thus be realized by the implementation of asuccession of instructions of a computer program product stored in thememory of a hard disk or on a removable support and running as follows:

-   -   choice of a meshing stepsize in particular as a function of the        wavelength of the main wave;    -   determination of the coordinates of the sources S_(i) and/or        S′_(i) and of the points of contact P_(i) and/or P′_(i);    -   choice of a type of wave involved and computation of the        coefficients of the interaction matrices applied to the points        P_(i) and/or P′_(i) through the implementation of matrix        computation software;    -   choice of a law of reflection and/or of transmission of the        surface of the obstacle and computation of the coefficients of        the reflection and/or transmission matrices;    -   computation of the values of the sources S_(i) and/or S′_(i);        and    -   computation of the scalar or vector quantities at every point of        three-dimensional space.

In this regard, the present invention is also aimed at such a computerprogram product, stored in a central unit memory or on a removablesupport able to cooperate with a reader of this central unit, andcomprising in particular instructions for implementing the methodaccording to the invention.

Of course, the present invention is not limited to the embodimentdescribed hereinabove by way of example; it extends to other variants.

Thus, it will be understood that, even if, in the figures discussedabove, both the surface of an obstacle and the surface of a radiatingelement are represented, the present invention applies also to theestimation of physical quantities within the framework of a waveinteracting with an obstacle and emitted in the far field. In thiscontext, it is not necessary to demarcate the surface of a radiatingelement to be meshed and relations [8] and [14] above suffice todetermine the interaction between this wave and the obstacle.

Equations making it possible to compute the scalar or vector quantitiesat a point M of space, for electromagnetic waves, or acoustic waves,have been indicated above. Of course, these quantities may be estimatedfor other types of waves, in particular for thermal waves,electromagnetic waves involving radiofrequency antennas, or others.

Of course, the present invention is not limited to an application tonondestructive testing, but to any type of application, in particular inmedical imaging, for example for the study of Microsystems employingacoustic microscopy with movable mirrors.

Interactions between a wave and a single obstacle have been describedabove. Of course, the present invention applies to an interaction withseveral obstacles. Accordingly, it is simply necessary to mesh thesurfaces of these obstacles and to add up their contribution for theestimation of a vector or scalar quantity at any point of space.Likewise, as indicated above, the surface of the obstacle OBS may beplane, or else curved, or else of any complex shape.

Thus, within the framework of a wave interacting with several obstaclesin space, a simulation equivalent to that represented in FIG. 5C wouldmake it possible to position sensors and/or radiating elements as afunction of the configuration of these obstacles, in particular for anapplication to the determination of the position of loudspeakers in apartitioned cabin, such as a motor vehicle cabin.

The three-dimensional space may be divided up into a plurality ofregions, as described above with reference to FIGS. 4A, 4B, 5A and 5B.However, in order for a surface of one of the said regions to beconsidered to be an obstacle of a main wave, the said obstacle becomingactive by emission of a secondary wave, the angle of incidence of themain wave on this surface must preferably remain less than or equal to90°.

1. Method of evaluating a physical quantity associated with aninteraction between a wave and an obstacle, in a region ofthree-dimensional space, wherein: a) a plurality of surface samples(dS_(i)), of which a part at least represents the surface of an obstaclereceiving a main wave and emitting, in response, a secondary wave, isdetermined by meshing, and at least one source (S_(i)) emitting anelementary wave representing a contribution to said secondary wave isallocated to each surface sample, b) a matrix system is formed,comprising: an invertible interaction matrix (F(M)), applied to a givenregion (M) of space and comprising a number of columns corresponding toa total number of sources, a first column matrix, each coefficient(v_(i)) of which is associated with a source (S_(i)) and characterizesthe elementary wave that it emits, and a second column matrix, which isobtained by multiplication of the first column matrix by the interactionmatrix and the coefficients of which are values of a physical quantity(V(M)) representative of the wave emitted by the set of sources in saidgiven region (M), c) to estimate the coefficients of the first columnmatrix (v_(i)), chosen values of physical quantity (V(P_(i))) areassigned to predetermined points (P_(i)), each associated with a surfacesample (dS_(i)), said chosen values (V(P_(i))) being placed in thesecond column matrix, and this second column matrix is multiplied by theinverse of the interaction matrix applied to said predetermined points(P_(i)), d) to evaluate said physical quantity (V(M)) representing thewave emitted by the set of sources in a given region (M) ofthree-dimensional space, the interaction matrix is applied to said givenregion (M) and this interaction matrix is multiplied by the first columnmatrix comprising the coefficients estimated in step c).
 2. Methodaccording to claim 1, wherein, to evaluate a physical quantityrepresentative of an interaction between an element radiating a mainwave and an obstacle receiving this main wave, in step a), a pluralityof surface samples (dS′_(i)) together representing an active surface ofthe element radiating the main wave is furthermore determined, bymeshing, and at least one source (S's) emitting an elementary waverepresenting a contribution to said main wave is allocated to eachsample of the active surface, steps b), c) and d) are furthermoreapplied to the samples of the active surface, and said physical quantity(V(M)) representing the interaction between the radiating element andthe obstacle in a given region (M) of three-dimensional space isevaluated by taking account of the contribution, in said given region(M), of the main wave emitted by the set of sources of the activesurface and the contribution of the secondary wave emitted by the set ofsources of the surface of the obstacles.
 3. Method according to claim 1,wherein each coefficient of the interaction matrix, applied to a givenregion of space, is representative of an interaction between a sourceand said given region and the value of each coefficient is dependent ona distance between a source and said given region.
 4. Method accordingto claim 1, wherein the interaction matrix applied, in step c), to saidpredetermined points (P_(i)) comprises a number of rows corresponding toa total number of predetermined points (P_(i)).
 5. Method according toclaim 1, wherein the physical quantity to be evaluated is a scalarquantity (V(P_(i))) and, in step a), a single source is allocated toeach surface sample.
 6. Method according to claim 5, wherein theinteraction matrix (F(M)) applied, in step d), to a region of space (M)comprises a row.
 7. Method according to claim 5, wherein eachpredetermined point (P_(i)), associated with a surface sample (dS_(i))corresponds to a point of contact between this surface sample (dS_(i))and a hemisphere whose surface is equal to the surface of this surfacesample, and whose centre corresponds to a position of the source (S_(i))which is allocated to this surface sample.
 8. Method according to claim5, wherein: the main wave is an electric wave, the coefficients of thefirst column matrix are values of electric charge that are eachassociated with a source, and the coefficients of the second columnmatrix are values of electric potential.
 9. Method according to claim 5,wherein: the main wave is a magnetic wave, the coefficients of the firstcolumn matrix are values of magnetic flux that are each associated witha source, and the coefficients of the second column matrix are values ofmagnetic potential.
 10. Method according to claim 5, wherein: the mainwave is a sound wave, the coefficients of the first column matrix arevalues of speed of sound that are each associated with a source, and thecoefficients of the second column matrix are values of acousticpressure.
 11. Method according to claim 1, wherein the physical quantityto be evaluated is a vector quantity (V(P_(i))) expressed by its threecoordinates in three-dimensional space, and three sources (SA_(i),SB_(i), SC_(i)) are allocated, in step a), to each surface sample(dS_(i)).
 12. Method according to claim 11, wherein the interactionmatrix (F_(v)(M)) applied, in step d), to a region of space (M)comprises a row for each space coordinate (X, Y, Z).
 13. Methodaccording to claim 11, wherein: the three sources allocated to eachsurface sample are substantially in one and the same plane, and eachpredetermined point (P_(i)) associated with a surface sample (dS_(i))corresponds to a point of contact between this sample and a hemispherewhose surface is equal to the surface of this sample, and whose centrecorresponds to the position of a barycentre of the three sources. 14.Method according to claim 13, wherein the three sources of one and thesame surface sample form substantially an equilateral triangle, and thetriangles of the surface samples are oriented substantially randomlywith respect to one another.
 15. Method according to claim 11, wherein:the main wave is an electric wave, the coefficients of the first columnmatrix are values of electric charge that are each associated with asource, and the coefficients of the second column matrix are values ofcoordinates of an electric field.
 16. Method according to claim 11,wherein: the main wave is a magnetic wave, the coefficients of the firstcolumn matrix are values of magnetic flux that are each associated witha source, and the coefficients of the second column matrix are values ofcoordinates of a magnetic field.
 17. Method according to claim 11,wherein: the main wave is a sound wave, the coefficients of the firstcolumn matrix are values of speed of sound that are each associated witha source, and the coefficients of the second column matrix are values ofcoordinates of an acoustic velocity.
 18. Method according to claim 1,wherein, to estimate the contribution of the secondary wave in saidgiven region in step d), said values of physical quantity (V(P_(i)))chosen in step c) are dependent on a predetermined coefficient ofreflection and/or of transmission of the main wave by each surfacesample of the obstacle.
 19. Method according to claim 18, taken incombination with claim 6, wherein the secondary wave corresponds to areflection of the main wave on the obstacle and the hemisphere isoriented outwards from the obstacle.
 20. Method according to claim 18,taken in combination with claim 6, wherein the secondary wavecorresponds to a transmission of the main wave in the obstacle and thehemisphere is oriented inwards into the obstacle.
 21. Method accordingto claim 19, wherein, in step c), the values (v′_(i)) associated withthe sources (S′_(i)) of the radiating element (ER) are determined and atleast the following are formulated: a first interaction matrix (F(P))representing the contribution of the sources of the obstacle to thepredetermined points of the surface of the obstacle (P_(i)), a secondinteraction matrix (F′(P)) representing the contribution of the sourcesof the radiating element to the predetermined points of the surface ofthe obstacle (P_(i)), a reflection (R) or transmission (T) matrix, whosecoefficients represent coefficients of reflection or of transmission ateach predetermined point (P_(i)) of the obstacle, to determine thevalues of the sources of the obstacle (v_(i)) as a function of thevalues of the sources of the radiating element (v′_(i)) and of amultiplication of the first and second interaction matrices and of thereflection or transmission matrix.
 22. Method according to claim 21,wherein, in step c), the values (v′_(i)) associated with the sources(S′_(i)) of the radiating element (ER) are determined by taking accountof the reception of the secondary wave by the radiating element (ER) andby furthermore formulating: a third interaction matrix (F(P′))representing the contribution of the sources of the obstacle to thepredetermined points of the surface of the radiating element (P′_(i)),and a fourth interaction matrix (F′(P′)) representing the contributionof the sources of the radiating element to the predetermined points ofthe surface of the radiating element (P′_(i)).
 23. Method accordingclaim 19, wherein the surface of the obstacle corresponds to aninterface between two distinct media of a heterostructure.
 24. Methodaccording to claim 1, wherein the main wave is a sound wave and thecoefficients of the interaction matrix are each dependent on an angle ofincidence of an elementary wave emanating from a source in said givenregion (M).
 25. Method according to claim 7, wherein, for each surfacesample, the value is tested of a scalar product of: a first vector({right arrow over (r)}) normal to the surface sample and directedtowards the apex (P) of the hemisphere (FIG. 7A), and a second vector({right arrow over (SM)}) drawn between a source (S) associated withthis hemisphere and said given region (M), while distinguishing: thecase where this scalar product is less than a predetermined thresholdand the contribution of this source is not taken into account, and thecase where this scalar product is greater than a predetermined thresholdand the contribution of this source is actually taken into account. 26.Method according to claim 1, wherein the main wave is a sound wave and,in step a), a total number of surface samples (dS_(i)) is chosensubstantially as a function of a wavelength of the sound wave so as tosatisfy the Rayleigh criterion.
 27. Method according to claim 1, whereina plurality of values of the physical quantity estimated in step d),which are obtained for a plurality of regions of space, are compared soas to select a candidate region for the placement of a radiating elementintended to interact with the obstacle.
 28. Method according to claim 2,wherein the radiating element is a sensor, for nondestructive testing,intended for analysing an object forming an obstacle of the main wave.29. Computer program product, stored in a central unit memory or on aremovable support able to cooperate with a reader of this central unit,wherein it comprises instructions for implementing a method ofevaluating a physical quantity associated with an interaction between awave and an obstacle, in a region of three-dimensional space, wherein:a) a plurality of surface samples (dS_(i)), of which a part at leastrepresents the surface of an obstacle receiving a main wave andemitting, in response, a secondary wave, is determined by meshing, andat least one source (S_(i)) emitting an elementary wave representing acontribution to said secondary wave is allocated to each surface sample,b) a matrix system is formed, comprising: an invertible interactionmatrix (F(M)), applied to a given region (M) of space and comprising anumber of columns corresponding to a total number of sources, a firstcolumn matrix, each coefficient (v_(i)) of which is associated with asource (S_(i)) and characterizes the elementary wave that it emits, anda second column matrix, which is obtained by multiplication of the firstcolumn matrix by the interaction matrix and the coefficients of whichare values of a physical quantity (V(M)) representative of the waveemitted by the set of sources in said given region (M), c) to estimatethe coefficients of the first column matrix (v_(i)), chosen values ofphysical quantity (V(P_(i))) are assigned to predetermined points(P_(i)), each associated with a surface sample (dS_(i)), said chosenvalues (V(P_(i))) being placed in the second column matrix, and thissecond column matrix is multiplied by the inverse of the interactionmatrix applied to said predetermined points (P_(i)), d) to evaluate saidphysical quantity (V(M)) representing the wave emitted by the set ofsources in a given region (M) of three-dimensional space, theinteraction matrix is applied to said given region (M) and thisinteraction matrix is multiplied by the first column matrix comprisingthe coefficients estimated in step c).